Question

Question about a book proof [topology]

Answer 1

My question is about the part where it talks about U_alpha being an indexed family of subsets. Wouldn't that oversimplify things? Can we actually assume that it is indexable? For example, Let's say we're talking about all the open intervals in (0, 1). You can't assign a U_1, U_2, etc. because there are uncountably infinite open intervals in (0, 1).

Sorry for the poor quality of the picture - it's dark...

Answer 2

*Talking specifically about example 3

Answer 3

So I take it if you have a problem with showing closure w.r.t. arbitrary union?

Answer 4

In order to show that \((X,\mathcal{T}_X)\) is a topology, you must show that \(X,\emptyset\in\mathcal{T}_X\) and it must be closed under arbitrary unions (I don't think it matters if it's countable or not) and finite intersections. So going back to your example of there being an uncountable number of subsets of (0,1), that's fine and all...but there are also an uncountable number of open subsets of (0,1) and it's possible to index them as \(\{U_{\alpha}\}_{\alpha \in A}\) where \(A\) is an uncountable index set. So when Munkres supposes that there's a family of \(\{U_{\alpha}\}_{\alpha\in A}\) with each \(U_{\alpha}\subseteq X\), we're just considering an arbitrary number of them (could be countable or uncountable).

I hope this clarifies things a little bit!

Answer 5

Ah, that's what I was missing. I was assuming the index alpha represented one of the positive integers. Thanks, this clarifies it completely!